The cutting stock problem with mixed objectives: Two heuristics based on dynamic programming

The purpose of this paper is to provide efficient methods for solving a large spectrum of industrial cutting stock problems. We propose two methods. Both are based on dynamic programming. In both methods, we reduce the computation burden by keeping, at each stage of the dynamic programming process, only the states which seem to be the most promising in terms of cost. One of the methods favours the computation time at the expense of the quality of the solution; this method is used in the sale department, where the goal is to propose a “good” solution in less than one minute. The second method favours the quality of the solution. Industrial applications are proposed.     

Limited memory discrete gradient bundle method for nonsmooth derivative-free optimization

Typically, practical nonsmooth optimization problems involve functions with hundreds of variables. Moreover, there are many practical problems where the computation of even one subgradient is either a difficult or an impossible task. In such cases derivative-free methods are the better (or only) choice since they do not use explicit computation of subgradients. However, these methods require a large number of function evaluations even for moderately large problems. In this article, we propose an efficient derivative-free limited memory discrete gradient bundle method for nonsmooth, possibly nonconvex optimization. The convergence of the proposed method is proved for locally Lipschitz continuous functions and the numerical experiments to be presented confirm the usability of the method especially for medium size and large-scale problems.     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

发展并行数值方法

1 引言 本世纪40年代中期至50年代初,第一台电子计算机和第一批存储程序计算机即vonNeumann计算机相继问世 。此后,计算机新陈代谢异常迅速,大约每隔5年运算速度增加10倍.50年代的计算机是串行结构,每一时刻只能按照一条指令对一个数据进行操作。由于电子信息传输速度以光速为极限,单靠改进线路已难于得到所期望的计算性能,串行计算机性能已接近了物理极限。为了克服传统计算机结构对提高运行速度的限制,从60年代起人们开始探索将并行性引入计算机结构设计,提出了研制并行计算机的设想。1972年单指令流多数据流并行计算机Illiac Ⅳ投入运行;1976年向量计算机Cray—1投入运行。在整个80年代,具有共享存储的并行向量计算机研制、生产和商售都获得了很大成功。当代高     

A new generalized compound Riccati equations rational expansion method to construct many new exact complexiton solutions of nonlinear evolution equations with symbolic computation

Based on symbolic computation and the idea of rational expansion method, a new generalized compound Riccati equations rational expansion method (GCRERE) is suggested to construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general complexiton solutions. The validity and reliability of the method is tested by its application to the (2+1)-dimensional Burgers equation. It is shown that more complexiton solutions can be found by this new method.     

Accelerated Imaginary-time Evolution Methods for the Computation of Solitary Waves

Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.     

Global Optimization using a Dynamical Systems Approach

We develop new algorithms for global optimization by combining well known branch and bound methods with multilevel subdivision techniques for the computation of invariant sets of dynamical systems. The basic idea is to view iteration schemes for local optimization problems – e.g. Newton’s method or conjugate gradient methods – as dynamical systems and to compute set coverings of their fixed points. The combination with bounding techniques allow for the computation of coverings of the global optima only. We show convergence of the new algorithms and present a particular implementation. Michael Dellnitz - Research of the authors is partially supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 376     

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.     

层次分析法权重计算方法分析及其应用研究

介绍层次分析法的基本概念,同时也分析了层次分析法权重的计算方法及应用,层次分析法的计算方法有四种方法:几何平均法、算术平均法、特征向量法、最小二乘法,以往的文献利用层次分析法解决实际问题时,都是采用其中的某一种方法求权重向量,不同的方法会导致结果有些偏差,将对一具体实例,采用四种计算方法分别得出权重向量再进行排序和综合分析,这样可以避免采用单一方法所产生的偏差,得出的结论将更全面、更有效。     

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.     

Unrestricted variables in linear programming

There are a number of ways of dealing with a linear programming problem in which some variables are allowed to take on negative values. One of these methods, which is either ignored or mentioned only incidentally in most textbooks, requires only one additional variable to be introduced regardless of how many unrestricted variables the original problem has. In this note, this method is interpreted geometrically and an application to the computation of extreme points is given.     

The splitting finite-difference time-domain methods for Maxwell's equations in two dimensions

In this paper, we consider splitting methods for Maxwell's equations in two dimensions. A new kind of splitting finite-difference time-domain methods on a staggered grid is developed. The corresponding schemes consist of only two stages for each time step, which are very simple in computation. The rigorous analysis of the schemes is given. By the energy method, it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions. Numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods. Furthermore, the methods are also applied to solve a scattering problem successfully.     

Partial fractions expansion: a review of computational methodology and efficiency

Nine methods for expressing a proper rational function in terms of partial fractions are presented for the case where the denominator polynomial has been reduced to linear factors. Only those methods which are amenable to computation algorithms are considered. To the extent possible, Newton's divided difference formula is used to provide a uniform derivational tool. Each method is illustrated numerically. The efficiency of the methods are compared on the basis of the number of multiplications and divisions required in the computation.     

Hybrid computation in mathematical programming

It is shown that hybrid computation facilitates the solving of problems in the field of mathematical programming. First, the most important features of hybrid computation are discussed. The main part of the paper deals with examples of computational methods in which hybrid computation can be applied succesfully. The main concentration is on constrained static parameter optimization, where for brevity only penalty function techniques are considered. The use of a penalty function having an extra parameter is discussed rather extensively.     

Projective interpolation of polynomial vectors and improved key recovery attack on SFLASH

SFLASH is an instance of the famous C* \(^{-}\) multivariate public key cryptographic schemes and it was chosen by the NESSIE cryptographic project of the European Consortium in 2003 as a candidate signature algorithm used for digital signatures on limited-resource devices. Recently, a successful private key recovery attack on SFLASH was proposed by Bouillaguet, Fouque and Macario-Rat by uncovering the kernel properties of quadratic forms of the central map. The most expensive step in the attack is the calculation of kernel vectors of skew-symmetric matrices over a bivariate polynomial ring. Bouillaguet et al. proposed two methods to accomplish this computation. Both methods involve symbolic computation on bivariate polynomials. The first method computes characteristic polynomials of matrices of polynomials and is very expensive. The second method involves a Gröbner basis computation and so its complexity is difficult to estimate. In this paper, we show this critical step of calculating kernel vectors can be done by numerical computation on field elements instead of symbolic computation. Our method uses a nondeterministic interpolation of polynomial vectors called projective interpolation, and its complexity can be explicitly evaluated. Experiments show that it is much faster, making the total attack on SFLASH about 30 times faster (the critical step is about 100 times faster) than the first method of Bouillaguet et al. The new method is also slighter faster than their second method.     

基于拉普拉斯变换有限差分方法的B-S期权定价

提供了一种基于自适应拉普拉斯变换有限差分方法来解决Black-Scholes期权定价问题.相比较于传统的时间推进法,此方法在保证较高精确度和很好的收敛性的同时,还可以减少计算时间.这一精确有效的方法将通过数值实验来验证.     

Convergence analysis of an optimally accurate frozen multi-level projected steepest descent iteration for solving inverse problems

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable.     

求解运输问题的一种新算法

本文将求解分派问题的标号算法成功地用于运输问题,并证明其中的非负处理可以省略,从而把Dijk-stra算法扩展到可能出现负边权的运输问题。与通常方法比较,这种方法具有直观、简单、计算量少、及易于推广等优点;最后证明该算法是多项式的,计算复杂性仅为o(n3)(当m≤n时)。     

质点法在SGARP中的新实现

可持续发展的几何自动推理平台(SGARP)支持用户发展多种多样基于规则的机器自动推理或人机交互推理方法,但缺乏处理符号计算的模块,其解题能力仍有待加强.质点法是最近发展的继面积法之后又一个能对可构造型几何命题生成可读机器证明的具有完全性的算法.基于一种在SGARP中快捷实现符号计算功能的方法,对质点法机器证明算法进行了新的实现.新添加的质点法模块使得用户能更便捷地验证更多的几何定理,从而使SGARP能更好地满足用户学习与发展几何机器推理的需求.     

A new solution method for state equations of nonlinear system

Along with the computation and analysis for nonlinear system being more and more involved in the fields such as automation control, electronic technique and electrical power system, the nonlinear theory has become quite a attractive field for academic research. In this paper, we derives the solutions for state equation of nonlinear system by using the inverse operator method (IOM) for the first time. The corresponding algorithm and the operator expression of the solutions is obtained. An actual computation example is given, giving a comparison between IOM and Runge-kutta method. It has been proved by our investigation that IOM has some distinct advantages over usual approximation methods in that it is computationally convenient, rapidly convergent, provides accurate solutions not requiring perturbation, linearization, or the massive computations inherent in discrietization methods such as finite differences. So the IOM provides an effective method for the solution of nonlinear system, is of potential application valuable in nonlinear computation.